We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).
相似文献The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.
相似文献Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.
相似文献We study the properties and applications of the directed graph, introduced by Hawkes in 1968, of a finite group \( G \). The vertex set of \( \Gamma_{H}(G) \) coincides with \( \pi(G) \) and \( (p,q) \) is an edge if and only if \( q\in\pi(G/O_{p^{\prime},p}(G)) \). In the language of properties of this graph we obtain commutation conditions for all \( p \)-elements with all \( r \)-elements of \( G \), where \( p \) and \( r \) are distinct primes. We estimate the nilpotence length of a solvable finite group in terms of subgraphs of its Hawkes graph. Given an integer \( n>1 \), we find conditions for reconstructing the Hawkes graph of a finite group \( G \) from the Hawkes graphs of its \( n \) pairwise nonconjugate maximal subgroups. Using these results, we obtain some new tests for the membership of a solvable finite group in the well-known saturated formations.
相似文献Consider the following nonparametric model: \(Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,\) where \(x_{ni}\in {\mathbb {A}}\) are the nonrandom design points and \({\mathbb {A}}\) is a compact set of \({\mathbb {R}}^{m}\) for some \(m\ge 1\), \(g(\cdot )\) is a real valued function defined on \({\mathbb {A}}\), and \(\varepsilon _{n1},\ldots ,\varepsilon _{nn}\) are \(\rho ^{-}\)-mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of \(g(\cdot )\). The rate can achieve nearly \(O(n^{-1/4})\) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.
相似文献Let \(K\subset {\mathbb {R}}^d\) be a bounded set with positive Lebesgue measure. Let \(\Lambda =M({\mathbb {Z}}^{2d})\) be a lattice in \({\mathbb {R}}^{2d}\) with density dens\((\Lambda )=1\). It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis for \(L^2({\mathbb {R}}^d)\) if and only if K tiles both by \(A({\mathbb {Z}}^d)\) and \(B^{-t}({\mathbb {Z}}^d)\). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis, then K can be written as a finite union of fundamental domains of \(A({{\mathbb {Z}}}^d)\) and at the same time, as a finite union of fundamental domains of \(B^{-t}({{\mathbb {Z}}}^d)\). If \(A^tB\) is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede’s type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.
相似文献We are concerned with the following \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\)
$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution \(u_{\lambda }\) of perturbation equations. First, we prove that \(u_{\lambda }\rightharpoonup u\) as \(\lambda \rightarrow 0\). Second, by using vanishing lemma, we get that \(u\) is a nontrivial solution of the original problem.
相似文献Given a prime \( p \) and a partition \( \sigma=\{\{p\},\{p\}^{\prime}\} \) of the set of all primes, we describe the structure of the nonnilpotent finite groups whose every Schmidt subgroup is \( \sigma \)-subnormal.
相似文献In this paper we study the following fractional Hamiltonian systems
$$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$where \(\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are the left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \(\mathbb {R}\) respectively, \(L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}\) and \(W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}\) are suitable functions. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.
相似文献We study the problem of recovering an unknown signal \({\varvec{x}}\) given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator \(\hat{\varvec{x}}^\mathrm{L}\) and a spectral estimator \(\hat{\varvec{x}}^\mathrm{s}\). The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\). At the heart of our analysis is the exact characterization of the empirical joint distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\), given the limiting distribution of the signal \({\varvec{x}}\). When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form \(\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}\) and we derive the optimal combination coefficient. In order to establish the limiting distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\), we design and analyze an approximate message passing algorithm whose iterates give \(\hat{\varvec{x}}^\mathrm{L}\) and approach \(\hat{\varvec{x}}^\mathrm{s}\). Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
相似文献We study integrals of the form
$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).
相似文献Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.
相似文献We consider the existence and uniqueness of solutions to initial value problems for general linear nonhomogeneous equations with several Riemann–Liouville fractional derivatives in Banach spaces. Considering the equation solved for the highest fractional derivative \( D^{\alpha}_{t} \), we introduce the concept of the defect \( m^{*} \) of a Cauchy type problem which determines the number of the zero initial conditions \( D^{\alpha-m+k}_{t}z(0)=0 \), \( k=0,1,\dots,m^{*}-1 \), necessary for the existence of the finite limits \( D^{\alpha-m+k}_{t}z(t) \) as \( t\to 0+ \) for all \( k=0,1,\dots,m-1 \). We show that the defect \( m^{*} \) is uniquely determined by the set of orders of the Riemann–Liouville fractional derivatives in the equation. Also we prove the unique solvability of the incomplete Cauchy problem \( D^{\alpha-m+k}_{t}z(0)=z_{k} \), \( k=m^{*},m^{*}+1,\dots,m-1 \), for the equation with bounded operator coefficients solved for the highest Riemann–Liouville derivative. The obtained result allowed us to investigate initial problems for a linear nonhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that the operator at the second highest order derivative is 0-bounded with respect to this operator, while the cases are distinguished that the fractional part of the order of the second derivative coincides or does not coincide with the fractional part of the order of the highest derivative. The results for equations in Banach spaces are used for the study of initial boundary value problems for a class of equations with several Riemann–Liouville time derivatives and polynomials in a selfadjoint elliptic differential operator of spatial variables.
相似文献This paper deals with the regularity of weak solutions to the 3D magneto-micropolar fluid equations in Besov spaces. It is shown that for \(0\le\alpha\le1\) if \(u\in L^{\frac{2}{1+\alpha}}(0,T; \dot{B}_{\infty,\infty}^{\alpha})\), then the weak solution \((u,\omega ,b)\) is regular on \((0,T]\).
相似文献In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula
$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).
相似文献